Nonsmooth finite-time stabilization of neural networks with discontinuous activations

This paper is concerned with the finite-time stabilization for a class of neural networks (NNs) with discontinuous activations. The purpose of the addressed problem is to design a discontinuous controller to stabilize the states of such neural networks in finite time. Unlike the previous works, such stabilization objective will be realized for neural networks when the activations and controllers are both discontinuous. Based on the famous finite-time stability theorem of nonlinear systems and nonsmooth analysis in mathematics, sufficient conditions are established to ensure the finite-time stability of the dynamics of NNs. Then, the upper bound of the settling time for stabilization can be estimated in two forms due to two different methods of proof. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design method.

[1]  V. Haimo.  Finite time controllers , 1986 .

[2]  Jinde Cao,et al.  On periodic solutions of neural networks via differential inclusions , 2009, Neural Networks.

[3]  M. Forti,et al.  Global convergence of neural networks with discontinuous neuron activations , 2003 .

[4]  Tianping Chen,et al.  Dynamical Behaviors of Delayed Neural Network Systems with Discontinuous Activation Functions , 2006 .

[5]  Edward Ott,et al.  Controlling chaos , 2006, Scholarpedia.

[6]  Yiguang Hong,et al.  Adaptive finite-time control of nonlinear systems with parametric uncertainty , 2006, IEEE Transactions on Automatic Control.

[7]  Quanmin Zhu,et al.  Nonsmooth Finite-Time Control of Uncertain Affine Planar Systems , 2006, 2006 6th World Congress on Intelligent Control and Automation.

[8]  Jinde Cao,et al.  Existence and Attractivity of Almost Periodic Solution for Recurrent Neural Networks with Unbounded Delays and Variable Coefficients , 2006 .

[9]  Zidong Wang,et al.  Exponential synchronization of stochastic delayed discrete-time complex networks , 2008 .

[10]  Jinde Cao,et al.  Exponential stability in the mean square for stochastic neural networks with mixed time-delays and Markovian jumping parameters , 2009 .

[11]  Jinde Cao,et al.  Exponential synchronization of stochastic perturbed chaotic delayed neural networks , 2007, Neurocomputing.

[12]  Ying-Cheng Lai,et al.  Controlling chaos , 1994 .

[13]  Ligang Wu,et al.  Exponential stabilization of switched stochastic dynamical networks , 2009 .

[14]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Eduardo Sontag.  A universal construction of Artstein's theorem on nonlinear stabilization , 1989 .

[16]  Yurong Liu,et al.  Stability criteria for periodic neural networks with discrete and distributed delays , 2007 .

[17]  Jinde Cao,et al.  Adaptive synchronization of uncertain dynamical networks with delayed coupling , 2008 .

[18]  Max Donath,et al.  American Control Conference , 1993 .

[19]  Long Wang,et al.  Finite-time formation control for multi-agent systems , 2009, Autom..

[20]  S. Bhat,et al.  Finite-time stability of homogeneous systems , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[21]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[22]  Wang Jin.  Existence and Attractivity of Almost Periodic Solutions of a Neural Network with Delays , 2002 .

[23]  L. Bushnell,et al.  Adaptive finite-time control of nonlinear systems , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[24]  Jorge Cortés,et al.  Finite-time convergent gradient flows with applications to network consensus , 2006, Autom..

[25]  Jinde Cao,et al.  Finite-time synchronization of coupled neural networks via discontinuous controllers , 2011, Cognitive Neurodynamics.

[26]  Chuandong Li,et al.  Exponential stabilization of chaotic systems with delay by periodically intermittent control. , 2007, Chaos.

[27]  Ju H. Park,et al.  New results on exponential passivity of neural networks with time-varying delays , 2012 .

[28]  Tianping Chen,et al.  Dynamical Behaviors of Delayed Neural Network Systems with Discontinuous Activation Functions , 2006, Neural Computation.

[29]  Dennis S. Bernstein,et al.  Finite-Time Stability of Continuous Autonomous Systems , 2000, SIAM J. Control. Optim..

[30]  Long Wang,et al.  Finite-Time Consensus Problems for Networks of Dynamic Agents , 2007, IEEE Transactions on Automatic Control.

[31]  Sanjay P. Bhat,et al.  Finite-Time Semistability and Consensus for Nonlinear Dynamical Networks , 2008, IEEE Transactions on Automatic Control.

[32]  Ju H. Park,et al.  New approaches on stability criteria for neural networks with interval time-varying delays , 2012, Appl. Math. Comput..

[33]  J. Hopfield,et al.  Computing with neural circuits: a model. , 1986, Science.

[34]  O. M. Kwona,et al.  Synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay , 2012 .

[35]  M. Forti,et al.  Global Convergence of Neural Networks With , 2003 .

[36]  J. Aubin,et al.  Existence of Solutions to Differential Inclusions , 1984 .

[37]  A. F. Filippov.  Equations with the Right-Hand Side Continuous in x and Discontinuous in t , 1988 .

[38]  Jinde Cao,et al.  Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches , 2011, Neural Networks.

[39]  Daniel W. C. Ho,et al.  Pinning Stabilization of Linearly Coupled Stochastic Neural Networks via Minimum Number of Controllers , 2009, IEEE Transactions on Neural Networks.

[40]  Frank L. Lewis,et al.  Finite-time distributed consensus via binary control protocols , 2011, Autom..

[41]  Jinde Cao,et al.  Exponential synchronization of chaotic Lur’e systems with delayed feedback control , 2009 .

[42]  Zhihong Man,et al.  Robust Finite-Time Consensus Tracking Algorithm for Multirobot Systems , 2009, IEEE/ASME Transactions on Mechatronics.

[43]  Tianping Chen,et al.  Almost Periodic Dynamics of a Class of Delayed Neural Networks with Discontinuous Activations , 2008, Neural Computation.

[44]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[45]  Jinde Cao,et al.  Local synchronization of one-to-one coupled neural networks with discontinuous activations , 2011, Cognitive Neurodynamics.

[46]  Yurong Liu,et al.  On delay-dependent robust exponential stability of stochastic neural networks with mixed time delays and Markovian switching , 2008 .